# Condorcet Method

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```					  Condorcet Method

Another group-ranking method
Development of Condorcet Method

 As we have seen, different methods of
determining a group ranking often give
different results.
 For this reason, the marquis de Condorcet,
a French mathematician, philospher, and
good friend of Jean-Charles de Borda,
proposed that any choice that could obtain a
majority over every choice should win.
Condorcet Method
 The Condorcet method requires that a
choice be able to defeat each of the other
choices in a one-on-one contest.
 Again consider the preference schedules
from the last section.
Condorcet Example
(CORRECTED)!
 We must compare each choice with every
other choice:

C         D
A         B

B          C        B         B

C          D        D         C

D          A        A          A
8         5        6          7
Condorcet Example (cont’d)
 Begin by comparing A with each of B, C and
D.
 A is ranked higher than B on 8 schedules
and lower on 18.
 Because A can not obtain a majority against
B, it is impossible for A to be the Condorcet
winner.
Choice B
 Next consider B.
 We have already seen that B is ranked higher than
A, so let’s now compare B with C.
 B s ranked higher on 8+5+7 = 20 schedules and
lower on 6.
 Now compare B with D. B is ranked higher on 8 +
5 + 6 = 19 schedules and lower on 7.
 B could be the Condorcet winner since it has a
majority over the other choices.
Tabular results
 The table on the left shows
A   B   C    D
all of the possible one-on-
one contests in this
A       L   L    L
example.
B   W       W    W    To see how a choice does
in one-on-one contests,
red across the row
C   W   L        W
associated with that
choice.
D   W   L   L
 Although the Condorcet method seems
ideal, it sometimes fails to produce a winner.
 Try this example!
You Try: Condorcet Example

A        B        C

B        C        A

C        A        B

20       20       20
Condorcet Example Explained
 Notice that A is preferred to B on 40 of the
60 schedules but the A is preferred to C on
only 20.
 For this reason, there
is no
Condorcet winner.
 You may expect that if A is preferred to B by
a majority of voters and if B is preferred to C
by a majority of voters, then a majority of
voters would prefer A to C.
 However, we have just seen that this is not
necessarily the case.
Transitive Property
 Do you remember the transitive property?
 If we consider the relation “greater than (>)”,
we see that if a > b and b>c, then a>c.
 However, according to the Condorcet method
the transitive property does not always hold
true.
 When this happens using the Condorcet
method, it is known as a Condorcet Paradox.
Extra Practice
1. The choices in the set of preferences
shown represent three bills to be
considered by a legislative body. The
members will debate two of the bills and
choose one of them. The chosen bill and
the third bill will then be debated and
another vote taken. Suppose you are
responsible for deciding which two will
appear on the agenda first.
Practice (cont’d)

A            B            C

B            C            A

A            B
C

40           30           30

 If you strongly prefer Bill C, which two bills
would you place first on the agenda? Why?
Practice (cont’d)

A             C        B           B
B             A        C           A

C             B        A           C
14        3          22
8

a. Use a runoff to determine the winner in the
set of preferences.

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